0 ----- 0 0 0 0 8 ]8 8 0 ]8 8

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1977) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

, Séria I: PRACE MATEMATYCZNE X X (1977)

M. K. B

(Kalyani, West Bengal)

Results on А С * — со (C-sense) and A C G * - со (C-sense) functions

Abstract.

In [2], absolutely continuous functions and generalized absolutely continuous functions in the restricted sense relative to to such as AC* — со and ACG* — to functions are defined. Following W .L .C . Sargent [9], we have introduced here further generalized functions such as AC* — to (C-sense) and ACG* — to (C-sense) {definitions

1.3, 1.4} and have studied some of their important properties.

1. Introduction. Let t»(x) be non-decreasing on the closed interval [a, b]. Outside the interval it is defined by со (ж) = со (a) for x < a and

<o(x) = co(b) for x > b. Let 8 and D denote respectively the set of points of continuity and discontinuity of co(x).

R. L. Jeffery [6] has defined the class °U of functions F(x) as follows:

F( x) is defined on [a, b]8 such that F{x) is continuous at each point of [ a , b ]8 with respect to the set 8. If x0eD, then F(x) tends to a limit {finite or infinite) as x tends to x0 + and x0 — over the points of the set 8.

These limits will be denoted by F ( x 0-f ) and F ( x 0 — ), respectively. When x < a, F( x) = F ( a Jr) and when x > b , F(x) = F ( b —). F(x) may or may not be defined at the points of the set D.

Let % denote the class of functions F(x) of Ш for which F (x Q-\-) and F (x 0 —) are finite, x0eD.

In [6], Prof. Jeffery has also defined co-derivatives of functions of the class °U and has defined Lebesgue-Stieltjes integral. In [5], this inte­

gral has been generalized to Perron-Stieltjes integral {definition 1.6}.

In [5], the following definition has been introduced:

D

1.1. Let a real function F(x) be defined finitely on [a,&]

and let it be Perron-Stieltjes integrable [or (PS)-integrable] on [a, b].

Write

ь

(co)C(F‘, a, b) — --- г (PS) Г ^F(x)dco.

co(o+ ) — co(a — ) J

a

F( x) is said to be Cesàro-continuous relative to со or (со) C-continuous at x0 if Lt (a>)C(F', xQ, X q -F h) = F ( x 0),

h—>0

8 М. К. Bose

(со) С (F; х0, x0 + h) =

x0+h

— ---- —r --- :--- г (PS) f ^F(t)dco, h > 0, co(x0 + J i ) - со(х0~ ) ф 0, co(xQ-\- h) — co(x0 — ) J

x0 x0+h

— — — — ---- — (PS) f F(t)dco, h < 0, co{x0 + h ) -ü )(x Q + ) # 0,

(Û\X

-\-}1) — C

( X

-\-) J

Xq

F ( x0 + h), a)(x0 + h) — ü)(x0= F )= 0 .

It is easily seen [5] that F(x) is (co)C-continuous at xeD. In [5] again the class %x of functions F( x) possessing the following properties is defined:

(i) F{x) is defined finitely on [a, b] such that F(x) is (PS)-inte- grable on [u, &];

(ii) at each point x0 of D, F(x) tends to a finite limit as x tends to х0 + or x0— over the points of 8 ;

(iii) at a point x0 of D , F (x) has the value where

(iv)

F ( X

+ ) + F (

0 - ) t

2 ’

1 ^1f(a) for x < a, F(b) for x > Ъ.

D

1.2 { [ 6 ] , definition 5.2}. Let F( x) eWx. For a point x c S and for h Ф 0 with x-{-he8, the function Ф(х, h) is defined by

Ф(х,к) —

(co)C(F; x, x + F) — F(x)

\{oo(x + h) — со (ж)} ’ 0,

со (x ф h) — со (x) Ф 0, co(x-\-h) — co(x) = 0.

The upper and lower limits of Ф(х, h) as 7&->0+ with x + heS are called the upper and lower Cesàro-derivates with respect to со [or upper and lower (со) C-dérivâtes'] of F (x) at x on the right and are denoted by CD +F w(x) and ÜD+F m(x) respectively. If ü D + F ^ x) = CD+JF(a(a?), the common value is called the (со) C-derivative of F( x) at x on the right and is denoted by CT>F+a(x). Similarly the left (oo) C-derivates СТ)~Жю(х), CD_ Ft 0(x) and the left (со) C-derivative 0D F _ a (x) of F(x) are defined.

The two sided upper and lower limits of Ф(х,Тг) as 0 are the upper

and lower (со) C-derivates of F(x) at x and are denoted by CDF^a?) and

CDF œ(x), respectively. If CDF m(x) = GDFm(x), the common value

is called the (со) C-derivative of F (x) at x and^is denoted by CDF^æ).

АС *— со and ACGr* — со functions Э4

We introduce the following definitions:

D

1.3. A function F ( x ) e ûU 1 is said to he AO* — œ (Cesàro sense) or briefly АС* —со (C-sense) over a set E c [a, 6] if to every posi­

tive number e there exists a positive number ô such that for any set o f non-overlapping open intervals {(cr, dr)} having end-points in E with

V (co(dr + ) — co(cr — )} < ô Г

the relations

Vbound|(co)0(F; cr, x) — F( er)\ < e r Cr<X<:dr

and

^ bound 1 ( со) G ( F ; dr, x ) —F(dr) | < e r cr<;X<df

hold.

D

1.4. A function F(x)e4tx is said to be ACG* —со (Cesàro>

sense) or briefly ACG* —со (C-sense) on [a, 6], if it is (со) C-continuous- on [a, 6] and if the interval can be expressed as the sum of a denumerable number of closed sets on each of which F(x) is AC* — со (C-sense).

The purpose of the present paper is

(i) to obtain a set of necessary and sufficient conditions for a function F(x) belonging to class 41 г to be AC* — со (C -sense) on a closed set Q <= [a, b]

and

(ii) to show that if a function F{x)*.4l1 is ACG* —со (C-sense) on [a, 6], then GDFw(x) exists finitely со-almost everywhere {definition 1.10}

in [a, b]8 and that this function becomes a constant on [a, &] if CDF M(x) vanishes со-almost everywhere on [a, b] 8 and if F ( x -f ) = F (x — ) for J e h .

We use the following notations:

N o ta tio n s .

$0 = 8X — blf Ct 27 &2> •••}»

i

S2 = 8SX, 8, = [a, b]8 - ( 80 + 8z),

where {(а{ , b{)} is the set of pairwise disjoint open intervals in [a, &] on each of which со (x) is constant. Further let S f and S f denote those points- of $ 2 which are respectively the left end-points and the right end-points of (ait bfj. (o*(E) denotes the outer co-measure {[3], [6]} of a set E while

\Е\Ш denotes the co-measure of the co-measurable [6] set E.

For a function F(x)e4t, the lower co-derivates [3] of F(x) on the

right and left at the point x will be denoted by B +F al(x) and D _ F m(x>

10 М. К. Бове

respectively. Further for the same function, the symbols F'+to(x), Р'_ш(х) and F'm(x) will denote respectively the co-derivative on the right, the

•со-derivative on the left and the co-derivative of F(x) at the point x.

We require the following known definitions:

D efinition 1.5 {[5], definition 2.1}. Let f {x) be a function (which is not necessarily finite) defined on the closed interval [«,&].

A function M(x) € is said to be a (PS)-major function of f(x) on [c*, £>] if (a) M(x) is non-decreasing on each open ftiterval <= [a, b], where со(x) is constant,

(b) M ( a - ) = 0,

(c) D _ M m{x) > — oo on $ 3 + $2 , D +M a{x) > — o o on S^ + S f , and

(d) M 'jx ) > f ( x ) on D , D__Mm(x) ^ f ( x ) on S3 + S~, D +M J x ) ^ f ( x ) on 83 + S£.

Analogously a (PS)-minor function is defined.

D efinition 1.6 {[5], definition 2.2}. A function f (x) is said to be integrable in Perron-Stieltjes sense relative to со [or (PS )-integrahle\ on

[a, b] if

(i) it has at least one (PS)-major function M(x) and at least one (PS)-minor function m(x) and

(ii) inf{Jf(& + )} = sup (m (& + )}•

If/(a ?)'is (PS)-integrable on [a, 6], the common value inf {31 (b + )}

= sup{m(&-f)} is called the Perron-Stieltjes integral [or {VS)-integral]

ь

of the function on [a , b ] and is denoted by (PS) f f{x)dco.

a

We require the following additional definitions:

D efinition |1.7. Let F { x ) e < ^ 1. For any x e S with x p h e S , the function W{x,h) is defined by

( Р { х р К ) — Р{х) ЧЦх, h) = co{x + Ji)-co{x) ’

l o ,

со (x + h) — со (x) t ^O, со (x + h) — со (x) — 0.

If W(x, h) tends to a limit as & ->0+, this limit is called the co-derivative

of F(x) at x on the right and is denoted by F'+0i{x). Similarly the left

co-derivative F'_m{x) is defined. If F'+C 0(x) = F'_(0{x), the common value

is called the co-derivative o î F { x ) at x and is denoted by F'i{x).

АС *—со and ACG*— со ftinctions 11

D

1.8. Let F( x )€% 1. For any x e S and a point £ {Ф oo) in 8 we define %{x, £) as follows:

*(®, £) =

F ( i ) - F { x )

co(tj) — 0)(x) ’ co(£) — o)(x) Ф 0, co(£) — co(x) — 0.

If #(#, £) tends to a limit as £ tends to ж over 8 except for a subset of 8 of co-density {[3], definition 3.1} zero at x, then the limit is the approxi­

mate co-derivative of F(x) at x and is denoted by {ap)F'w{x).

D

1.9. If a function F(x)e°Ux is such that for every posi­

tive number e there is a positive number ô such that for any set of non­

overlapping open intervals {(cr, dr)} having end-points on a set E a [a, 6]

for which

У {a)(dr + ) — co(cr — )j < ô

Лтш*

r

we h a v e.

\F(dr) - F ( c r)\ < e,

then we say that F (x ) is AC — со on F.

D

1.10. If a property P is satisfied at all points of a set A except on a set of co-measure zero, then it is said that P is satisfied со-almost everywhere in A.

In the sequal we shall require the following theorems :

T

1.1 {[5], Theorem 3.1}. I f a function f(x) is defined and (PS)-integrable on [a , b], then the indefinite Perron-8tieltjes integral of f(x) belongs to the class <Ш 0.

T

1.2 {[2], Theorem 4.5}. Let a function f (x) defined on [a , b]

be summable (LS) [3] over a closed set Q <=. [a ,b ] with end-points c, d and complementary intervals {{cn, d n)} and let f (x) be (PS)-integrable on each closed interval [cn, dn1. I f

______ P

On = bound I (PS) f f{x)day I cn<ct</?<dn a

and

% О п < о о , П

then f (x) is (PS)-integrable on the closed interval [c, d] and

d dn

(PS) j f ( x ) d o > = ( LS) //(® )d w + j T { ( P S ) f f ( x ) d < o - W n},

c Q n cn

12 М. К. Бове

Wn = f ( c n) [со{сп + ) - ( о { е п- ) ] + f ( d n) [co(dn + ) - œ { d n- ) ] .

We have followed [1], [3] and [6] for terms and notations not men­

tioned earlier.

2. Preliminary lemmas.

L

2.1. I f a function F ( x )€^1 is АС* — со (C-sense) on a set E cz [a, b], then it is АС —со on E.

P r o o f. The lemma follows from the definition of АС* — со (C-sense) and the inequalities

\F(dr) - F ( c r)\

< l(«>) c (F) <V, dr) ~ F { c r)\ + |(co)C(P; cr, dr) — F{dr)\

< bound I(co)C(F-, cr, x) — F ( e r)\ -fbound |(со)О(Р; dr, x) — F(dr)],

cr < x < d r cr*Zx<dr

where (er, dr) is an open interval having end-points in E.

L

2.2. I f a function F(x) is in class <%0, then F ( x ± ) is bounded on [a, 6].

P r o o f. W e first suppose that F(x) is not bounded on [a, b]S. It is then possible to get a sequence of points x 1} x 2, ..., xn, ... of [a, b]S such that

\F(xn)\ > Mn,

where M 1 < M 2 < ... is a sequence of positive numbers with J fn—>oo.

Since the sequence х х, х г, . . . is bounded, it has a limit point x (say).

Then x e[a, b] and F { x ± ) becomes infinite. This is a contradiction. Hence F( x) is bounded on [a , b]S. Now let x be any point of the interval [a, &], then corresponding to this point x we can get a sequence of points Ц , £2, ...

of [u, b]S of which x is the limit. If M and m are respectively the upper and lower bound of F (x ) on [a, b]S, then

m < F ( £ n) < M . Taking limit, we get

m < F ( x ± ) ^ i M . It follows that F ( x ± ) is bounded on [a, 6].

L

2.3. I f a function F (х)€.Шг is AC — со on a closed set Q c [a, &], then it is bounded on Q.

P r o o f. Since F(x) is АС —со on Q, it follows that for every e > 0, there exists a <5 > 0 such that

£ \F(dr) - F (cr)\ < e

where

АС *—со and ACG* — со functions 13

for any set of non-overlapping open intervals {(cr, dr)} having end-points in Q snch that

JT1 {a){dr + ) - c o { c r — )} < <3.

From this it follows that if xeQS is a limit point of Q from the right or left, then

F(x) = F ( x + ) or F ( x - ) .

lit now follows as in Lemma 2.2 that F (x) is bounded on Q.

L

2A. I f a function F ( x ) e %1 is (to) C-continuous on an interval [c, d] c [a , ft], then (со) G ( F; c, x) and (со) G(F\ d , x) are bounded on (c , d]

and [c, d) respectively.

P r o o f. Write

and

\ (со) G(F-, c, x) î o y x > c , H(x) = {

( F(c) for x < c;

0

Ж (PS) J ^F(*)dco

a

F (& -)

for x < a, for a < x < 6, for x > b.

W e distinguish the following cases:

C a se (i). For every x > с, оо(х-\-) — co(c —) ^ 0.

In this case

Thus

H (x)

G(x + ) - G ( c ~ ) со (x + ) — со (c — ) F(c)

for X > c , for x < c .

H(x) =

G ( x ) - G ( c ~ ) co(x) — c o ( c - ) H ( x + ) F(c)

for Xe(c, d ] S , for xe(c, d]D, for X = c.

B y Theorem 1.1, G(x)eW0 and so H(x)e<%0. So by Lemma 2.2, R ( x ± ) is bounded on [c, d]. Hence И (x) is bounded on [c, d].

Case (ii). For a certain a e (c,d ], co(a + ) — co(c—) = 0 and co(x-f ) —

— c o ( c - ) Ф 0 for x > a.

14

Here

M. К. Bose

Н(х) =

G(x + ) - G ( c - ) со(а? + ) — (о (с — ) F(x)

for х > а, for х < а.

(со) C-contimiity of F(x) implies continuity of F(x) on [c, a] and so F( x) is bounded on [c, a], i.e., H(x) is bounded on [c, a]. Therefore S ( x ) is bounded on [c, d] in case a — d. If a < d, then as in case (i), H (x) is bounded on [a, d]. Therefore H(x) is bounded on [c, d].

It now follows that (со) C( F- ,c, x) is bounded on (c, d]. Similarly we can show that (со) G (F-, d, x) is bounded on [c, d).

L emma 2.5. I f a function F ( x ) e c^1 is АС* —со (C-sense) on a closed set Q a [ a, b] having end-points c, d and (со) C-continuous on [c, d], then for any set of non-overlapping open intervals {(cr, dr)} having end-points in Q

^bound|(co) G(F; cr, x) — F( cr)\ < oo

r cr < x < d r

and

J^bound](co) G(F; dr, x ) —F(dr)\ < oo.

у*

P r o o f. Since F(x) is АС* —со (C-sense) on Q, there exists a <5 > 0 such that for any set of non-oyerlapping open intervals {(xi , х'{)} haying end-points in Q for which

{co(a?J + ) — co(x{ — )} < ô

i

we have

^ bound I ( со) C(F; xi} x) —F ( x i) | < 1

i Х {< х< Х{

and

^ b o u n d |(co) C(F; x\, x) — F ( x i)\ < 1.

i xi<Lx<xi

We consider the following cases:

Case (i). The saltus of co(x) at every point of [c, d] is less than <5/2.

In this case we can break up the interval [c, d] into finite number of subintervals [p0, P i l lPi, P2] , • • •, , P J (G = Po < Pi < • • • < Pn = à ) such that

(!) oo(pi + ) - o o ( p i_ l - ) < 5/2 >

i = 1, 2 , . . . , n. Let {(cr, dr)} be any set of non-overlapping open intervals having end-points in [рг_ ! , р г] Q. Then by (1)

JT1 (co(dr + ) - c o ( c r- ) } < <5.

15- АС* —со and ACG* — a> functions

Therefore

bound|(co) G(F; cr, x) — F( cr)\ < 1

r cr<x<:dr

and

У ] bonnd[(ft)) G(F; dr, x ) —F(dr)\ < 1.

r cr^x<dr

Now let {(xr, yr)} be any set of non-overlapping open intervals having end-points in [PoiPïlQ- И all the intervals lie in [p0,Pil, then clearly

Vbound|(co) G(F-, xr, x) — F ( x r)\ < 1

r . xr< x*iyr

and

^ bound I (a») G(F; yr, x ) —F ( y r)\ < 1.

If the intervals {(xr, yr)} lie in both [p0, p x] and [px, p 2] let m be the integer such that xm is less than p x and ym is greater than p x. Since F(x) is (со) C-continuous on [c, d], by Lemma 2.4 (со) G (F) xm, x) and (со) G ( F ; y m, x) are bounded on (xm, ym] and [xm, ym) respectively. Since F(x) is АС* —со (C-sense) on Q it follows from Lemma 2.1 and Lemma 2.3 that F(x) is- bounded on Q. Let the positive number M be such that

|(co) G{F; xm, x)\ < M for xe{xm, ym],

!(co) G{F; ym, x ) \ ^ M for xe\xm, y m), and

\F(x) \ < M for XeQ.

Now

bound I { œ ) C { F ; x m, x ) - F { x J \

xm<x^Vm _____

< bound (|(co) G(F; xm, x)\ + \F{xm)\} < 2M . xm<x^vm

Similarly

bound !(co) G{F] ym, x ) - F ( y m)\^ 2M.

хт^х<Ут Therefore

bound |(co) G(F-, xr, x) — F { x r)\

f Уф

= y j bound \{co)G{F-,xr, x ) - F { x r)\ + bound |(co) G { F ‘, x r, x) — F ( x r)\ -b

f<mXr<X^yr r>mxr<x^yr

+ bound J(co) G(F', xm, x) — F ( x m)\ < l + l-f-2 ilf = 2(1 + M).

16 М. К. Bose

Similarly

V bound |( со) C(F-,yr, æ) — F ( y r)| < 2(1 + I f).

r xr^x<yr

Lastly, in case there exists an integer m' such that xm. = p x or ym, = p lt then clearly

V b ou n d [(со) C { F ; x rJ x) — F ( x r)\ < 2

r xr< x < y r

and

bound [(со) C(F; yr, x ) - F ( y r)\ < 2.

r xr< x < y r

Therefore from the above considerations we see that ‘the lemma is proved in this case.

Case (ii). There are points in [c, d] at which the saltns of со (a?) > <5/2.

The points at which the saltus of a>(x) + <5 /2 are finite in number.

We denote these points by tl f t2, . . . t tq, where ^ < t 2 < ... < t q. In R - и hh we take points a, (a < ^) of 8 such that

<

2

со (a) — со + ) < <5 /2 , to (t{ — ) ~ со (ft) < (3/2.

Then at each point of [a, ft], the saltus of co(x) is less than <5/2 and there­

fore by case (i) the given property is satisfied in [a, ft]Q. Now let {(lr, mr)}

be any set of non-overlapping open intervals having end-points in [/г_ х, a]Q.

Proceeding as in case (i) we see

bound \(co)C(F; llf x) — P(Zi)| < 2M , where M' is a suitable positive number. Again by (2)

£ {(o{mr + ) - œ ( l r - ) } < ô.

r > 1

Therefore

and so

Similarly

2 ; bound I(co)C(P; lr, x) — F( lr)\ < 1

r > l V

l„<x^m.

^ b o u n d \(co)C(F‘, lr, x) — F{lr)\ < 1 + 2M ' . l~<x^m

^ b o u n d \(co)G(F; mr, x) — F{mr)\ < 1 + 2Ж', l~^x<m

It follows that the given property holds in [^_u a]Q. Similarly it can be

shown that the property holds in [ft, t{]Q. So proceeding as in case (i)

we easily see that the above property holds in Q. This proves the lemma.

А С *—со and АС G* — со functions 17

L emma 2.6. I f a function F { x ) e dU 1 is bounded on a closed set Q a [a ,b ] having complementary intervals {(си, dn)} and if

2 bound \(m)C(F-,cn, x ) - F { c n)\ < oo,

n cn<X^ dn

2 bound \(a))C(F‘,d n, x ) - F ( d n)\ < oo,

n Cn ^ X < ' d Tl

then

У

2 bound |(PS) J F{t)d(oI < oo.

n c n < x < V ^ d n X

P r o o f. Let the number M be such that

\F(x)\ < M for all xeQ and let

X

G(x) = (PS) J F(t)dco.

a

Let cn < x < dn and co(x + ) — co(cn — ) =£0; then

X

\в(х + ) - в ( с п-)\ = j(PS) f F(t)dm

c n

1

*

< {o)(dn + ) - œ { c n- j } ,--- --- - j(PS) f F{ t) dJ w(a? + ) — (o{cn — ) I J I

en

= {o>(dn + ) - ( o { c n-)}\{w)C{F-,cn,x)\

< {co{b + ) - a ) ( a - ) } \ ( ( o ) C ( F ‘, cn, x ) - F { c )l)| +

+ i^n ~Ь ) — co(cn — ) } M . Next let cn < x ^ dn and со(ж + ) — co(cn — ) = 0; then

X

\G(xJr ) —G{c.n — )\ = |(PS) J F(t)doo\ = 0.

cn

Therefore,

bound \G(x-\-)—G{cn—)\

cn< x < d n

< {co(b + ) - « ( < * - ) } bound \(a))C{F’,cn, x ) - F ( c n)\ + {a>(dn + ) - (o(cn- ) } 3 I

cn < x < sdn 2

2 — Roczniki PTM — Prace Matematyczne X X

18 М. К. Bos*

and so

(3) ^ bound \G(x + ) — G(cn — )\

n Cn<X^ dn

^ { ( o ( b + ) - c o ( a - ) } £ bound \(a>)C(F’, cni x ) - F ( c n)\ +

11 Cn<~X^ d1l

+ [ ^ { m ( d n + ) - o >( c n- ) } ] M

n

< {co(b + ) — co(a — ) } ^ bound |(co)C(F', cn, x ) —F( cn)\-f-

n cn< x < dn

- j - 2 -£co (Ô -f-) — со (a — 'ŸfJÿL < o o r

Similarly we can prove

(4) bound \G(oc — ) — G(dn + )\ < oo.

n cn ^ X<dn

Now,

________

У

bound |(PS) J F(t)dco\

cn< x < y < a n x

= bound \G(y + ) — G(x — )]

cn^Zx<y<dn

< bound \G(dn + ) — G(x — )\ + bound \G(y + ) — G(cn — )| +

cn^ x < d n Cn < v ^ d n

+ \G{dnJr) — G(cn — )\

and therefore by (3) and (4)

________

V

У bound |(PS) f F(t)dco\< oo.

n cn^ x < y ^ dn x

3. Theorems.

T

3.1. Let Q <= [a, b] be a closed set having end -points c, d

and complementary intervals {(cn, dn)}. The necessary and sufficient condi­

tions for a function F( x) belonging to class and (со) C-continuous on [c , d]

to be AO* — со (C-sense) on Q are that (i) F(x) is AO — coonQ, (ii) V bound \(co) 0 (F-, cn, x ) - F ( c n)\ < oo n cn < x ^ d n

and

У bound \(co)C(F-, dn, x) — F(dn)\ < oo, n cn ^ x < d n

and (iii) F(x) is constant on [a, jS] whenever co(j5 + ) — со (a — ) = 0 (a, fteQ).

АС* — со and ACGr* — со functions 19

P r o o f. Let e > 0 be chosen arbitrarily. To prove that the conditions are necessary, let us assume that F(x) is АС* —со (C-sense) on Q. There exists therefore a positive number rj such that for any set of non-overlap­

ping open intervals {(xi , xf)} having end-points in Q with

we have

and

If for a, fieQ,

{x>{xi + ) - ( o { x i - ) } < rj

i

^ bound I(co)G(F; x i} x) — F ( x {)\ < e i x^<x^x^

J ? bound |(co)C(.F; х{ , x) — F(x'i)\ < e.

г x^x<x^

co(/? + ) — co(a —) = 0 <r ), then from above it follows that

bound \(co)G(F; a, x) — F(a)\ < s

a<æ</3

and

bound |(co)0(-F; p, x ) - F ( P ) \ < e.

а<Ж<)3

Since s > 0 is arbitrary, it follows that

bound \(o))C{F-, a, x) — F(a)\ = bound \(co)G(F, ft, x) — F(P)\ = 0

a<£c</3 a < x < P

and so F(x) is constant in [a, P] which is condition (iii). From Lemma 2.1 and Lemma 2.5 we get that conditions (i) and (ii) are necessary.

To prove that the conditions are sufficient, we proceed as follows:

Since F(x) is AC — со on Q, it follows that F (x) is continuous at points of QS with respect to the set Q. Since the set D is atmost enumerable, F(x) is co-measureable on Q. Also by Lemma 2.3, F(x) is bounded on Q.

Hence F(x) is summable (LS) on Q. Again since F(x) is АС —со on Q, there exists a positive number <5 such that

JT \F{xi) - F { x i)\ < e/8

г

for any set of non-overlapping open intervals {(жг-, #'■)} with end-points in Q for which

£ {«>(»< + ) — « ( » < — )} < <5-

20 М. К. Bose

Then it follows that

(5) JT1 bound |JT(^) — F { x {)\ < e/8.

i Xl<t^X.

UQ

W e find a positive integer 1c such that

Let

= min[(5, co(dn-{-) — co{cn — ) (w < & and co(dn + ) — co(cn — ) =£0)]

and let {(pr, qr)} be any set of non-overlapping open intervals having end-points in Q such that

£ M 2 r + ) - w ( p r - ) } < <5j.

We see that none of the intervals {pr, qr) contain any of the intervals (cx, di), (ck1 dk) except possibly those for which o)(dn-\-) — co(en — ) = 0.

Let p r < x < qr. Then {co)C(F-,pr, x) — F ( p r)

X

--- -(P S ) f [ F ( t ) - F ( p r)]da> if co(x + ) ~ co{pr - ) # 0,

= со (a? 4-) — <*>\Pr ~ ) J

Pr

F ( x ) —F ( p r) if со{ x - f ) — oo(pr — ) — 0.

Case (i). If m(x-\-) — ю(рг — ) Ф 0, then by Lemma 2.6 and The­

orem 1.2.

( 8 ) ( o) ) C{ F; pr, x ) - F ( p r)

1

о(ж + ) — to(pr ^{— г (LS)} j l F ( t ) - F ( p r№ <0 +

+ I

1

со(ж + ) — co(pr — ' (PS) [ F ( t ) - F ( p r)ldcoF

+ - O)(0D-

X

Л

— г ---r (PS) f [ F ( t ) - F ( p r) ] dc o -

— CO { P r ) ' j

АС* — со and ACG* — со functions 21

2r

^{--- :}

{[F{ en) - F { p r))[co{cn + ) - m { c n- )' } + to{x A t ) — ы{Рг — )

+ [F(dn) - F (ргП [co(dn + ) - c o ( d n- ) ] } ~ 1 [^ (cs) - ^ ( 3 ? r)][co(c8 + )-« > (c s - ) ] ,

co(x-\-) — C o (p r ~ )

where Q denotes the common part of Q and the interval [pr, x] and

X

denotes the summation over n for which p r < cn < dn < x (n > k) and the third and fifth terms are absent unless x lies in the complementary interval {cs, ds). Now

(9) со (ж + ) — со { p r — ) (LS) J [ F ( t ) - F ( p r)-]dm

co(x + ) — co(pr — ) bound \ F {t)-F (p r)\\Q\t UQ

< bound IF (i) — F (pr) I.

UQ

X

If p r < en < dn < x and co(dn+ ) — co(cn —) Ф 0, then dT l

( 10 ) 1

o){x + ) — co{pr ~ ) _______ 1_______

со(х + ) — со(рг — )

(PS) J [.F(J)-*■ (}> ,)]*»

(PS) f [F(t) — F( cn)]dco +

+ \ F ( c „ ) - F ( p r) I

°>(<*« + ) - « > ( e» - ) — --- -(P S ) f

— CO(C„— )

J

cn

+ \F(cH) - F ( p r)\

\(co)C(F-,cn, d n) - F ( c n)\ + \F(cn) - F ( p

со (dn + ) — ы(сп — ) co(x + ) — co{pr — )

+

со {dn-r-) — (o (cn—)

CO {X -f* ) — CO ( P r — )

w(^n+) — coiGn ~ ) . co{x + ) — co(pr — ) and if co(dn -\-) — to (c n — ) = 0, then

(11) co(x ф) — co(pr — ) (PS) J [ F ( t ) ~ F ( p r)]dm = 0,

22 М. К. Bose

If p r < cs < x < ds, then

(

12

co(x + ) — a>(pr —) (PS) [ F ( t ) - F ( p r)]dco

_ a>(x4-) — < d ( cv — )

< \(co)G(F;cs, x ) - F ( c s)\ + \F(cs) - F ( p r)\ ---}JL-L . co(x-\-) — (o{pr — ) Combining the results of (8), (9), (10), (11) and (12) we get

(13) \(co)G(F-,pr, x ) - F ( p r)\

< bound \(a))C{F-,cn, x ) - F ( c n)\ + \(co)G{F-,cs, x ) - F ( c s)\ +

сп<х<Лп

+ 4 bound IF ( t ) —F ( p r)\.

UQ

Case (ii) If a)(x + ) — co(pr — ) = 0 and if X€(cs, ds)’, then

\(co)C(F‘, p r, x ) - F ( p r)\ = \ F ( x ) - F ( p r)\

< \ F ( x ) - F ( c s)\ + \F(cs) - F ( p r)\

= \(co)G{F8‘, cs, x) — F( c s)\ (by condition (iii)).

Therefore for any x in p r < x < qr, where oi(x-\-) — <x>{pr — ) = 0 , (14) \(co)C(F] p r, x) — F ( p r)\ < \{(o)G(F’, c3, x ) - F ( c s)|.

Therefore from (13) and (14)

bound \((o)C(F-,pr, x ) - F ( p r)\^ ^ bound \(w)G(F; cn, x ) - F ( c n)| +

r Pr<x<iqr n > k cn< x ^ d n

+ 4 V b o u n d \ F ( t ) - F ( p r)\ <^~ + -^ = s [by (5) and (6)].

r 2 2

Similarly we can show

bound \( ü >)G(F; qr, x) — F(qr)\ < s.

r pr<x<qr

Therefore F( x) is АС* — со (C-sense) on Q. This completes the proof of the theorem.

N o te 3.1. Before we prove the next theorem we observe that the concept of АС —о introduced by Jeffery {[6], definition 1; [4]} on a set F cz [a, b]S for a function belonging to class aU is identical with the con­

cept of АО — со for a function belonging to class Щх. Secondly, we observe

that the concept of BY — со introduced in papers [1] and [4] on a set

E cz S

for a function belonging to class °U is identical with the concept

АС *—со and ACG*—со functions 23

of B Y {[7], p. 118; [8], p. 215}. Hence, if a function belonging to class °UX be АС — со on a closed set Q <= S3, then {[1], cf. Theorem 5} it is BY on Q.

We also see that the definitions of со-dérivative [6] and of approxi­

mate co-derivative {[4], definition 1.2} of a function belonging to class % at points of the set S coincide with the definitions of the respective deri­

vatives introduced in this paper for a function belonging to class %x.

Arguing along standard ways {[7], cf. Theorem 5.14} it can be shown that if a function F{x)c°Ux be BY on [a, b], then F'œ(x) exists finitely co-almost everywhere in [a , Ь]$ {[1]; cf. Theorem 1; [3], cf. Theorem 6.2} and that if a function F { x ) e ûU1 be BY on a со-measurable set E c: [a, ft], then (ap)F'm{x) exists finitely со-almost everywhere in ES {[4], cf. Theorem 3.5}.

In Lemma 7.1 [5] Dutta has shown that for a function F( x) eW 0 with

and

Е { х ^ ) 1-Е{х{ - ) F ( x {) = ---,

CDF _ J x ) = F f j x ) for x e S3 + S~

C D F . J x ) = F'+(0{x) for ooe83 + 8.

provided the right-hand members exist. The same result holds in case

F (

)

<% 1 . In fact the proof holds verbatim. We state the result in the form of a lemma:

L emma 3.1. I f the function F(x) belongs to class then CB F _ J x ) = ^ _ ы{х) for xe S3 + S f and

CT>F+aj(x) = F'+a{x) for x e S 3 + S+

provided the right-hand members exist.

T heorem 3.2. Let a function F { x ) e aU1 be {со) C -continuous on [a, b].

I f it is AC* — со (C-sense) on a oo-measurable set E c= [a, 6], then C DF (a{x) exists finitely co-almost everywhere in ES. Also CD F m(x) is equal to (ар) F m(x) со-almost everywhere in ES.

P r o o f. Let A = E S - ( S 0 + S2). Then A is со-measurable subset of S3.

Let Q be a closed set contained in A such that

\QL ^ l-^-l ^со

where e is an arbitrarily chosen positive number. Let [c, d] be the smallest closed interval containing the set Q. Denote the component intervals of the open set [c, d] — Q by {(en, dn)}. By Theorem 3.1

^ bound \{oo)G{F; cn, x ) - F { e n)\ <

(15) oo

24 М. К. Bose

and (16) Let

£ bound \(a>)C(F;dn, x ) - F ( d n)\< oo.

n d'n

M n = bound (o)<7(P;cn,æ).

We introduce the function Jf(a?) as follows:

F(x) for X e Q ,

Жп for a?€(cH,d „), w = 1, 2 , . . . , U(c) for x < c,

F(d) for x > d.

Using (15) we get

(1 7 ) £ \M n - X ( c n)\ < o o.

n

Since F(x) is АС* — со (C-sense) on Q, we get by Lemma 2.1 that F(x) is АС—со o n # and so by note 3.1, F(x) is BY on Q. Using (17) it now follows that M(x) is B Y on [a, &] and so bounded on [a, &]. Again since F(x) is АС —со on #, it is continuous at points of Q with respect to the closed set Q. So F{ x) is bounded and co-measurable on Q and F(x) is the­

refore summable (LS) on Q. It follows that M(x) is w-measurable on [a, 6].

Since 31 (x) is bounded on [a , b], M(x) is summable (LS) on [u, &]. It now follows that M{x) belongs to class So, by note 3.1,

(18) 3 f j x ) = ( a p ) ^ (x),

со-almost everywhere in Q. Now let a be a point of Q. Then for h > 0 with аЗ-JieS we get by Lemma 2.6 and Theorem 1.2,

a-rh dn a+7t

(PS) j F{x)do> = (LS) f F(x)dco+ £ (PS) f F(x)dco-f (PS) f F(x)dco,

a

Q

h

where Q denotes the common part of Q and the interval [a, a + h],

h

denotes the summation over n for which a < cn < dn < a -f h and the last term is absent unless a + h lies in a complementary interval (cq, dq).

Since for c{ < x < dt,

X

(PS) J F(x)dco = [ft)(a? + ) — со(с{ — )](со)(7(Р; с{х) ci

X

< \a>{x + ) - ( o { c i -)']Mi = (LS) J M(x)d(of

ч

А С *—со and ACG* — со functions 25

we get

a + h a + h

(PS) j F{ x) dc o^ (LS) J M(x)dco-{- ^ (LS) J M(x)dco + (LS) J M{x)do>

« q c n Cq

a + h

— (LS) J M (x)dco.

Therefore

(co)C(F; a, a + й)— -F(a) (co)C(M', a, а-\-Ь,) — М(а) co(a + &) — (o(a) " со (a-\-h) — со (a) Taking limit as with a-f-heS, we get

(19) CD+ F „(а) < C D + JfJa).

If

mn = bound (co)C{F-, cn, x) Cn<'X^‘^"n

and if m(x) be a function defined by

m„ for ooe(cn, dn), n = 1 , 2 , M (x ) elsewhere,

m(x) =

and if we proceed with m{x) in the above manner we can show that m(x) is BY on [a , fe] and is in class °UX. Further

(20) m'a(x) = (ap)F'a(x),

co-almost everywhere in Q and

(21) CD+F t0(a) > CD+w <0(a), aeQ.

If

P n = bound (co)C(F; dn, x)

cn ^ x < d n

and

p% = bound {œ)C{F-, dn, x)

P(x) =

cn^ x < d n

and if P(x) and p(x) be two functions defined by

Pn

for xe{cn, dn), n = 1, 2, M (x) elsewhere ;

and

\pn for X€{cn, dn), n = 1, 2, p(x) = {

\ M(x) elsewhere,

we get using (16) that P(x) and p(x) are BY on [a, b] and are in class %x.

26 М. К. Bose

Further

(

2 2

Р'Лх) = К ( » ) = (ар)А'и(<в)>

co-almost everywhere in Q and

(24)

(23) O D - F J a ) ^ G D - P n{a), aeQ,

СТ>__Рш(а) > С В _ р ю(а), aeQ.

Since each of M(x), m(x ), P( x) and p(x) belongs to class °UX, we get by Lemma 3.1, and the seven relations (18)-(24) that CDF w{x) exists finitely ш-almost everywhere in Q and equals (%rp)F'a{x). Now since iSf]^ — 0 and since e > 0 is chosen arbitrarly, it follows that at со-almost all points of ES, C D F œ(x) exists finitely and equals (ap)F^(a?). This completes the proof of the theorem.

From the above theorem we can state following theorem:

T

3.3. I f a function F ( x ) e W1 is ACG-* — со (C-sense) on [a, ft], then CDF^ix) exists finitely w-dlmost everywhere in [a, b]S. Also C D F0)(x) is equal to (ap)F^(a?) co-almost everywhere in [a, b]S.

T

3.4. I f a function F { x ) e aU 1 is ACG* —со (C-sense) on [a, b]

amd CDF a(x) — 0 co-almost everywhere in [ a , b] S and if F(x-\~) = F ( x —) for a 7el), then F(x) is constant on [a, ft].

P r o o f . Since F( x) is ACG* —со (C-sense) on [a, 6], it follows that [a, ft] = £ Q ni where the sets Qn are closed and F(x) is АС* —со (C-sense) on each Qn. Let К be the set of points of [a, ft] throughout no neighbourhood of which F (æ) is constant. Then A is a closed set. Now let (c, d) be a comple­

mentary interval of K. Then F(x) being (со) C-continuous on \c, d], it is constant on \c, d]. Thus К has no isolated point and so it is a perfect set. If possible, let К be not null. It then follows from Baires theorem that there is an integer n and an interval [p, q] such that K [ p , q] is not nmpty, K [ p , q] = Qn[p, q] — T (say). The set T is then closed and F(x) is АС* —со (C-sense) on T.

Now we shall show that F(x) is АС —со on [a, /5], where a, (} are the end-points of T. Since F( x) is АС* —со (C-sense) on T, it follows from Lemma 2.1 that F(x) is АС —со on T and so for every positive number e there is a positive number <5 such that for any set of non-overlapping open intervals {(xi , xf)} having end-points in T for which

£ {со(х1 + ) - с о ( х { - ) } < ô we have

I F ( X i ) - F ( x {)\ < s. '

А С *—со and ACG* — со functions 27

Let {{cr, dr)} be any set of non-overlapping open intervals on [a, /?] snch that

{co(dr + ) — co(cr — ) } < ô .

If there are points of T on \cr, dr~\, let cr be the first point of T to the right of cr and let cr = cr if creT. Let dr be the first point of T to the left of dr and let dr = dr if dreT. Since F(x) is constant on [cr, cr] and on [dr, dr]

we have

F (c r) ~ F ( c r) = F ( d r) - F ( d r) = 0 . Now

(25) у IF(dr) - F ( c r)\

r

^ y \ F ( c r) - F [ c r) \ + V \ F ( d r) - F ( d r)\ + ] ? I F ( d r) - F ( c r)\

r T r

= У \F(dr) - F ( c r) \ < e ,

r

since

^ {co(dr + ) ~ co(cr — )} < ô.

Г

Thus F(x) is АС —со on f a , fi].

From (25) it follows that

and Write

F(x) = F ( x - f ) for x e [ a , P ) $ , F(x) ~ F { x — ) for xe(a, fi]>S■

F ( x ) G (x) = F (a) -F(fi)

for Же [a, /3], for x < a, for x > ft.

Then G(x) belongs to class ^ 0. Since F(x) is АС —со on [a, /1] and F(x)

= F ( x f - ) = F ( x — ) for xeD, it follows that G(x) is АС —со on [a, /3].

So, by'Theorem 1 [6] and Lemma 7.1 [5], G(x) is constant on [a, /?].

That is F(x) is constant on [a, /5] which contradicts the fact that a, (3 are end-points of T which is a portion of K. Thus К is null. So every point of [a,b] is interior to an interval on which F(x) is constant. By Heine- Borel theorem there is a finite number of intervals covering [a, b] on each of which F( x) is constant. Because F(x) is (со) C-continuous on [a, 6], it follows that F(x) is constant on [a, b].

In conclusion, I express my sincere gratitude to Dr. M. C. Chakra-

barty for his kind help and suggestions in the preparation of the paper.

28 М. К. Bose

References

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[2] M. K . B o s e , On Perfon-Stieltjes and Denjoy-Stieltjes integral, Communicated to J. Aust. Math. Soc. for publication.

[3] M. C. C h a k r a b a r t y , Some results on co-derivatives and B Y — со functions, J. Aust.

Math. Soc. Yol I X , parts 3 and 4 (1969), p. 345-360.

[4] — On functions of generalized bounded co-variation, Fund. Math. 66 (1970), p. 293-300.

[5] D . K . D u t t a , Perron-Stieltjes integral and Cesàro-Perron-Stieltjes integral, Colioq. Math, (to appear).

[6] R. L. J e f f e r y , Generalized integrals with respect to functions of bounded varia­

tion, Canad. J. Math. 10 (1958), p. 617-628.

[7] — The theory of functions of a real variable, Toronto 1962.

[8] I. P. N a t a n s o n , Theory of functions of a real variable, Vol I, New York 1955.

[9] W . L . C. S a r g e n t , A descriptive definition of Cesaro-Perron integrals, Proc.

London Math. Soc. 47 (1941), p. 212-247.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF KALYANI WEST BENGAL, INDIA